1.7 Eigen

1.8 Cayley-Hamilton

\[ \lvert {\bf A} - {\bf A} {\bf I} \rvert = 0 \]

Thus, you can substitute $\bf A$ in it's own characteristic equation.
This can be used to reduce the order of a matrix polynomial by using
the characteristic equation to find the value of a high order matrix
power as a linear combination of lower order powers of that matrix.

6.1 Pole Placement

The idea is to pick eigenvalues, $λ$, giving the desired
behavior, then determine a gain matrix $\bf K$ that will give the
controlled system those eigenvalues. This is done by solving the
characteristic equation for $\bf K$. This gives you one scalar
equation for each $λ$.

\[ \left| (\bf A - \bf B \bf K) - λ \bf I \right| = 0 \]

An identical but computationally easier approach is to solve this
equation by matching the proper coefficients:

\[ \left| (\bf A - \bf B \bf K) - λ \bf I \right| =
∏j=1n(λ - λj) \]

6.2 Optimal Control

We define some metric that our controller must minimize. For the LQ
problem, we minimize:

Where ${\bf e}$ is the error. The derivation is $\dot{\bf e}$ is
simple algebra. Thus, the problem of observation reduces to the
problem of control. Note that the poles for ${\bf A} - {\bf L} {\bf
C}$ can essentially be arbitrarily fast as the observer system only
exists inside a computer. In practice, observer poles should be about
10 times faster than controller poles.

7.3 Separation Principle

The separation principle states that one may design a feedback
controller assuming real $\bf x$ and an observer to find $\hat{\bf x}$
independently, then feed $\hat{\bf x}$ to the controller and have
everything still work. This is explained by: